Analysis of Error Metrics of Different Levels of Compression on Modified Haar Wavelet Transform

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 1, January 2011
1

Analysis of Error Metrics of Different Levels of
Compression on Modified Haar Wavelet
Transform
1
T.Arumuga Maria Devi,Assistant Professor,Centre for Information Technology and
Engineering,Manonmaniam Sundaranar University,Tirunelveli .TamilNadu.

2
S.S.Vinsley, Student IEEE Member,Guest Lecturer,Manonamaniam Sundaranar University,Centre
for Information Technology and Engg,Manonmaniam Sundaranar University, Tirunelveli.TamilNadu.

contains a large amount of spatial redundancy in plain areas
where adjacent picture elements (pixels, pels) have almost
Abstract—Lossy compression techniques are more efficient in same values. It means that the pixel values are highly
terms of storage and transmission needs. In the case of Lossy correlated [31]. The basic measure for the performance of a
compression, image characteristics are usually preserved in the
compression algorithm is Compression Ratio (CR). In a lossy
coefficients of the domain space in which the original image is
transformed. For transforming the original image, a simple but compression scheme, the image compression algorithm
efficient wavelet transform used for image compression is called should achieve a tradeoff between compression ratio and
Haar Wavelet Transform. The goal of this paper is to achieve image quality [32]. The balance of the paper is organized as
high compression ratio in images using 2 D Haar Wavelet follows: In section II describes the properties and advantages
Trnasform by applying different compression thresholds for the of haar wavelet transformation. In section III considers the
wavelet coefficients and these results are obtained in fraction of
seconds and thus to improve the quality of the reconstructed procedure for haar wavelet transformation. In section IV,
image ie., to arrive at an approximation of our original image. Implementation methodologies for wavelet compression and
Another approach for lossy compression is, instead of linear algebra are discussed. In section V describes the
transforming the whole image, to separately apply the same algorithm for its implementation. In section VI considers the
transformation to the regions of interest in which the image comparison for metrics. In section VII considers the graphs,
could be devided according to a predetermined characteristic.
HWT image compression output results, and these results are
The Objective of the paper deals to get the coefficients is nearly
closer to zero. More specifically, the aim of the thesis is to plotted between various parameters. In section VIII
exploit the correlation characteristics of the wavelet coefficients elaborates on the importance of this paper, some applications
as well as second order characteristics of images in the design of and its extensions. Quality and compression can also vary
improved lossy compression systems for medical images. Here a according to input image characteristics and content. Images
modified simple but efficient calculation schema for Haar need not be reproduced exactly. An approximation of the
Wavelet Transform.
original image is enough for most purposes, as long as the
Index Terms—Haar Wavelet Transform – Linear Algebra error between the original and the compressed image is
Technique – Lossy Compression Technique – MRI tolerable. Lossy compression technique can be used in this
area.

I. INTRODUCTION
II. PROPERTIES AND ADVANTAGES OF HAAR WAVELET
TRANSFORM
I Nrecent years, many studies have been made on wavelets.
An excellent overview of what wavelet have brought to the The Properties of the Haar Transform are described as
follows:
fields as diverse as biomedical applications, wireless
communication, computer graphics or turbulence [30]. Image
compression is one of the most visible applications of  Haar Transform is real and orthogonal. Therefore
wavelets. The rapid increase in the range and use of Hr=Hr* (1)
electronic imaging justifies attention of systematic design of Hr-1 = Hr T (2)
an image compression system and for providing the image Haar Transform is a very fast transform .
quality needed in different applications. A typical still image

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 The basis vectors of the Haar matrix are sequency IV. IMPLEMENTATION METHODOLOGY
ordered.
 Haar Transform has poor energy compaction for Each image [27] is presented mathematically by a matrix
images. of numbers. Haar wavelet uses a method for manipulating the
 Orthogonality: The original signal is split into a low matrices called averaging and differencing. Entire row of a
and a high frequency part, and filters enabling the image matrix is taken, then do the averaging and differencing
splitting without duplicating information are said to be process. After we treated entire each row of an image matrix,
orthogonal. then do the averaging and differencing process for the entire
 Linear Phase: To obtain linear phase, symmetric filters each column of the image matrix. Then consider this matrix
would have to be used. is known an semifinal matrix (T) whose rows and columns
 Compact support: The magnitude response of the filter have been treated. This procedure is called wavelet
should be exactly zero outside the frequency range transform.
covered by the transform. If this property is satisfied, Then compare the original matrix and last matrix
the transform is energy invariant. that is semifinal matrix(T), the data has became smaller.
 Perfect reconstruction: If the input signal is Since the data has become smaller, it is easy to transform and
transformed and inversely transformed using a set of store the information. The important one is that the treated
weighted basis functions, and the reproduced sample information is reversible. To explain the reversing process
values are identical to those of the input signal, the we need linear algebra. Using linear algebra is to maximize
transform is said to have the perfect reconstruction compression while maintaining a suitable level of detail.
property. If, in addition no information redundancy is
present in the sampled signal, the wavelet transform is, A. Wavelet Compression Methodology
as stated above, ortho normal.
From Semi final Matrix (T) is ready to be compressed. [29]
No wavelets can possess all these properties, so the choice Definition of Wavelet Compression is fix a non negative
of the wavelet is decided based on the consideration of which threshold value ε and decree that any detail coefficient in the
of the above points are important for a particular application. wavelet transformed data whose magnitude is less than or
Haar-wavelet, Daubechies-wavelets and biorthogonal- equal to zero (this leads to a relatively sparse matrix). Then
wavelets are popular choices [1]. These wavelets have rebuild an approximation of the original data using this
properties which cover the requirements for a range of doctored version of the wavelet transformed data. In the case
applications. of image data, we can throw out a sizable proportion of the
detail coefficients in this and obtain visually acceptable
The advantages of Haar Wavelet transform as follows: results. This process is called lossless compression. When no
information is loss (eg., if =0). Otherwise it is referred to as
1. Best performance in terms of computation time. lossy compression (in which case  >0). In the former case,
2. Computation speed is high. we can get our original data back, and in the latter we can
3. Simplicity build an approximation of it. Because we know this, we can
4. HWT is efficient compression method. eliminate some information from our matrix and still be
5. It is memory efficient, since it can be calculated inplace capable of attaining a fairly good approximation of our
without a temporary array. original matrix. Doing this, we take threshold value ε=10
ie., reset to zero all elements of semifinal matrix(T) which are
less than or equal to 10 in absolute value. From this we obtain
III. PROCEDURE FOR HAAR WAVELET TRANSFORM the doctored matrix. Then apply the inverse wavelet
transform to doctored matrix we get the reconstructed
To calculate the Haar transform of an array of n samples: approximation R.
In this process, we can get a good approximation of
1. Find the average of each pair of samples. (n/2 the original image. We have lost some of the detail in the
averages) image but it is so minimal that the loss would not be
2. Find the difference between each average and the samples noticeable in most cases.
it was calculated from. (n/2 differences)
3. Fill the first half of the array with averages.
4. Fill the second half of the array with differences.
5. Repeat the process on the first half of the array. (The array
length should be a power of two)

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B. Linear Algebra Methodology  Display the resulting images and comment on the
quality of the images.
To apply the averaging and differencing using linear
 Calculate MSE, MAE & PSNR values of different
algebra [27] we can use matrices such as A1,A2,A3…..An.
Compression Ratios for corresponding Reconstructed
that perform each of the steps of the averaging and images.
differencing process.
 Then add a small amount of white noise to the input
image. Default : variance =0.01, sigma=0.1, mean=0
i. When multiplying the string by the first matrix of the
first half of columns are taking the average of each pair  To compute the haar wavelet transform, set all the
and the last half of columns take the corresponding approximation coefficients to zero except those
differences. whose magnitude is larger than 3 sigma.

ii. The second matrix works in much the same way, the first  This same case is applicable to detail coefficients
half of columns now perform the averaging and that is horizontal, vertical & diagonal coefficients.
differencing to the remaining pairs, and the identity  Reconstruct an estimate of the original image by
matrix in the last half of columns carry down the detail applying the corresponding inverse transform.
coefficient from step i.
 Display and compare the results by computing the
iii. Similarly in the final step, the averaging and differencing root mean square error, PSNR, and mean absolute
is done by the first two columns of the matrix, and the error of the noisy image and the denoising image.
identity matrix carries down the detail coefficient from
 The same process is repeated for various images and
previous step.
compare its performance.
iv. To simplify this process, we can multiply these matrices
together to obtain a single transform matrix Alternative approach Algorithm is described as follows:
W=A1A2A3) we can now multiply our original string by
just one transform matrix to go directly form the original 1. Read the image cameraman.tif from the user.
string to the final results of step iii.
2. Using 2D wavelet decomposition with respect to a haar
v. In the following equation we simplify this process of wavelet computes the approximation coefficients matrix
matrix multiplication. First the averaging and CA and detail coefficient matrixes CH, CV, CD
differencing and second the inverse of those operation. (horizontal, vertical & diagonal respectively) which is
obtained by wavelet decomposition of the input matrix
ie., im_input.
1. T= ((AW)T W))T
T= (WT AT W)T 3. From this, again using 2D wavelet decomposition with
respect to a haar wavelet computes the approximation
T= WT (AT)T (WT)T
and detail coefficients which are obtained by wavelet
T=WTAW (3)
decomposition of the CA matrix. This is considered as
level 2.
2. (W1)-1 T W-1 = A
(W-1)1TW-1=A (4) 4. Again apply the haar wavelet transform from CA matrix
which is considered as CA1 for level 3.
5. Do the same process and considered as CA2 for level 4
V. ALGORITHM 6. Take inverse transform for level 1, level 2, level 3 &
level 4 that ie., im_input, CA, CA1, CA2.
 Read the image from the user.
7. Reconstruct the images for level 1, level 2, level 3 &
 Apply 2 D DWT using haar wavelet over the image level 4.
 For the computation of haar wavelet transform, set 8. Display the results of reconstruction 1, reconstruction 2,
the threshold value 25%, 10%, 5%, 1% ie., set all reconstruction 3, reconstruction 4 ie., level 1 , 2 , 3 , 4
the coefficients to zero except for the largest in with respect to the original image.
magnitude 25%, 10%, 5%, 1% . And reconstruct an
approximation to the original image by apply the
corresponding inverse transform with only modified
approximation coefficients.
 This simulates the process of compressing by factors
of ¼,1/10,1/20,1/100 respectively.

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VI. METRICS FOR COMPARISON  
 2 
PSNR=10xlog10  255  (9)
A. Compression Metrics  ! N1 N 2 2 
  ( f ( x, y)  g ( x, y)) 
The type of compression metrics [21] used for digital data  N1 N 2 y 1 z 1 
varies quite markedly depending on the type of compression And also Mean Absolute Error is denoted by
employed that use a simple ratio formula as given below

1 N
Output File size (bytes) MAE= |g(x,y)-f(x,y)| (10)
CompressionRatio(%)= x100 (5) N i 1
Input File size (bytes)
The standard error measures a large distortion, but the
A measure of rate is far more suitable metric for large
image has merely been brightened. Individually, MSE, SNR
compression ratios as it gives the number of bits per pixel
and PSNR are not very good at measuring subjective image
used to encode an image rather than some abstract
quality, but used together these error metrics are at least
percentage.
adequate at determining if an image is reproduced at a certain
8 x Output File size (bytes) quality.
Rate(bpp)= (6)
Input File size (bytes)
Rate, for image coding purposes, uses bits per pixel (bpp) VII. RESULTS AND DISCUSSIONS
as its unit.
The project deals with the implementation of the haar
B. Error Metrics wavelet compression techniques and a comparison over
various input images. We first look in to results of wavelet
In general, error measurements are used on lossy compression technique by calculating their comparison ratios
compressed images to try and quantify the quality of a and then compare their results based on the error metrics
picture. Getting a quantifiable measure of the distortion which is shown in Table I.
between two images is very important as one can try and Table I
minimize this thesis so as to better replicate the original Different types of Error Metrics with respect to Various
image. There are many ways of measuring the fidelity of a Compression Ratios.
picture g(x,y) to its original f(x,y). One of the simplest and Compression Error Metrics
most popular methods is to use the difference between f and Ratios MSE RMSE PSNR
g. In its most basic form is the mean square error (MSE) [11] 4:01 9937.92 99.6891 18.7841
given by, 10:01 14380.1 119.917 15.0893
1 N1 N2 20:01 15990.1 126.452 14.028
MSE=
N1 N 2
  ( f ( x, y)  g ( x, y))
y 1 z 1
2
(7) 100:1 17453.28 132.1109 13.1524

where f and g are N1xN2 size image. This is a very useful A. Effects on Compression Ratio Vs Various Parameters
measure as it gives an average value of the energy loss in the
lossy compression of the original image f. Signal-to-noise Wavelet Compression is applied for all images and the
ratio (SNR) is another measure often used to compare the compression ratio is being calculated. The image quality thus
performance of reproduced images which is defined by, measured in compression techniques is compared using a
BAR CHART, which proves the image quality of the various
 ! N1 N 2  input images which reconstructed images are better. This is

SNR=10xlog10 
 f ( x, y) 2
N1 N 2 y 1 z 1

(8) shown in Fig 1 & 2.
 COMPRESSION RATIO VS MSE
 ! N1 N 2 2 
  ( f ( x, y)  g ( x, y)) 
 N1 N 2 y 1 z 1  50000
40000
4:01
30000
MSE

10:01
SNR is measured in dB's and gives a good indication of the 20000
20:01
ratio of signal to noise reproduction. This is a very similar 10000 100:01:00
measurement to MSE. A more subjective qualitative 0
measurement of distortion is the Peak Signal-to-noise ratio
if
if

tif
f
f

.ti
.ti

r.t
t.t

.
ce
ri
an

ar
gh

(PSNR) [3].
m

ri
cm

m
ei

ne
bo

IMAGES
Fig. 1. Effects on Compression Ratio Vs MSE with Respect to Various
Input Images

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PSNR values depend on image type and cannot be used to
COMPRESSION RATIO Vs PSNR compare images with different content.
70

60
4:01
The quality of the reconstructed image is measured using
50 the error metrics:
10:01
40
 MSE
PSNR

30 20:01
 PSNR
20 100:01:00
10
The values of MSE & PSNR are calculated for all test images
0
and are provided in Table II.
cman.tif mri.tif eight.tif bonemarr.tif rice.tif

IMAGES
Table II
Fig. 2. Effects on Compression Ratio Vs PSNR with Calculation of different kinds of Error Metrics with respect to
Respect to various input images Various Input images under Compression Ratio 100:1

In the first experiment, we report on Compressed Ratio No of
consumed by Error Metrics as described in the previous SI.No Image elements to MSE PSNR
section. In our experiments, we used the gray scale sample, store in bytes
cameraman.tif of size 256*256. And measured the 1 cman.tif 256*256 17543.280 13.1524
compression ratio and the PSNR of the compressed image. In 2 Rice.tif 256*256 13094.653 16.0257
each case, the threshold level was changed and got the output. 3 Mri.tif 128*128 1248.827 39.5257
The results are presented in Fig 5. The x-axis represents 4 Eight.tif 242*308 41227.182 4.5567
Compression Ratio, while the y-axes represent MSE with 5 Bonemarr.tif 238*270 28009.786 8.4222
various input images.

In the Second experiment, we report on Compression Ratio VIII. CONCLUSION
Vs PSNR with various input images. In our experiments, we
used the gray scale image samples. In each case, the threshold This paper reported is aimed at developing computationally
level was changed and got the output. The results are efficient and effective algorithm for lossy image compression
presented in Fig 5. The x-axis represents Compression ratio, using wavelet techniques. This paper is particularly targeted
while the y-axes represents PSNR with various input towards wavelet image compression using Haar
images. Transformation with an idea to minimize the computational
requirements to achieve good reproduction image quality. In
The quality of compressed image depends on the no of this direction the following methods are developed. This
decompositions. The no of decompositions determines the image compression schemes for images have been presented
resolution of the lowest level in wavelet domain. Using larger based on the 2 D HWT. The promising results obtained
no of decompositions, that will be more successful in concerning reconstructed image quality as well as
resolving important HWT coefficients from less important preservation of significant image details, while on the
coefficients. After decomposing the image and representing otherhand achieving high compression rates.
it with wavelet coefficients, compression can be performed by
ignoring all coefficients below some threshold. In this  High compression ratio and better image quality
experiment, compression is obtained by wavelet coefficient obtained which is better than existing methods.
thresholding. All coefficients below some threshold are
 In addition, the above methods are to be for noisy
neglected and Compression Ratio is computed. Compression
images
Algorithm operation is follows : Compression Ratio is fixed
to the required level and threshold value has been changed to  To improve the quality of the reconstructed image
achieve required Compression Ratio after that PSNR is  The results are executed in fraction of seconds.
computed. PSNR tends to saturate for a larger no of
 To obtain the wavelet coefficients are nearly closer to
decompositions. For each compression ratio, the PSNR
zero.
characteristic has “threshold” which represents the optimal
no of decompositions. Below and above the threshold PSNR  Image denoising techniques will allow precise imaging
decreases and no of decomposition increases. PSNR is at much faster rates by greatly reducing the necessary
increased up to some no of decompositions. Beyond that, averaging time to construct low noise images.
increasing the no of decomposition has a negative effect.

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The performance of an image compression algorithm is implementation issues such as bit allocation methods and
basically evaluated in terms of compression ratio, MSE, and error estimation can be studied.
PSNR. A `good' algorithm has a high compression ratio.
Wavelet based image compression theory has rapidly Image denoising method using wavelet for noisy image
increased in the last seven years. With the increasing use of could be developed. This yield better result in image
multimedia technologies image compression requires higher compression techniques using wavelet for noisy input images.
performance as well as new functionality. To address this
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Fig. 3: Original and Haar Wavelet Transformed images for Different Levels

Fig. 4: Reconstructed Images for different Levels

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